Similarity, Congruence, and Proofs.notebookfloydmodelhigh.sharpschool.net/UserFiles/Servers/Server... · Similarity, Congruence, and Proofs.notebook 3 August 10, 2015 Jul 1912:33

Similarity, Congruence, and Proofs.notebook

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August 10, 2015

Jul 1911:26 AM

IntroductionThink about resizing a window on your computer screen. You can stretch it vertically, horizontally, or at the corner so that it stretches both horizontally and vertically at the same time. These are nonrigid motions. Nonrigid motions are transformations done to a figure that change the figure’s shape and/or size. These are in contrast to rigid motions, which are transformations to a figure that maintain the figure’s shape and size, or its segment lengths and angle measures. Specifically, we are going to study nonrigid motions of dilations. Dilations are transformations in which a figure is either enlarged or reduced by a scale factor in relation to a center point.

Lesson 1.1.1

Jul 1911:29 AM

Key Concepts• Dilations require a center of dilation and a scale factor.• The center of dilation is the point about which all points are stretched or compressed.

• The scale factor of a figure is a multiple of the lengths of the sides from one figure to the transformed figure.

• Side lengths are changed according to the scale factor, k.• The scale factor can be found by finding the distances of the sides of the preimage in relation to the image.

• Use a ratio of corresponding sides to find the scale factor:

• The scale factor, k, takes a point P and moves it along a line in relation to the center so that .

Jul 1911:38 AM Jul 1911:38 AM

Key Concepts, continued• If the scale factor is greater than 1, the figure is stretched or made larger and is called an enlargement. (A transformation in which a figure becomes larger is also called a stretch.)

• If the scale factor is between 0 and 1, the figure is compressed or made smaller and is called a reduction. (A transformation in which a figure becomes smaller is also called a compression.)

• If the scale factor is equal to 1, the preimage and image are congruent. This is called a congruency transformation.

• Angle measures are preserved in dilations.• The orientation is also preserved.• The sides of the preimage are parallel to the corresponding sides of the image.

• The corresponding sides are the sides of two figures that lie in the same position relative to the figures.

Jul 1911:39 AM

Key Concepts, continued• In transformations, the corresponding sides are the preimage and image sides, so and are corresponding sides and so on.

• The notation of a dilation in the coordinate plane is given by Dk(x, y) = (kx, ky). The scale factor is multiplied by each coordinate in the ordered pair.

• The center of dilation is usually the origin, (0, 0).• If a segment of the figure being dilated passes through the center of dilation, then the image segment will lie on the same line as the preimage segment. All other segments of the image will be parallel to the corresponding preimage segments.

• The corresponding points in the preimage and image are collinear points, meaning they lie on the same line, with the center of dilation.

Jul 1911:40 AM

Properties of Dilations

1.Shape, orientation, and angles are preserved. 2.All sides change by a single scale factor, k. 3.The corresponding preimage and image sides are parallel. 4.The corresponding points of the figure are collinear with the center of dilation.

Common Errors/Misconceptions• forgetting to check the ratio of all sides from the image to the preimage in determining if a dilation has occurred

• inconsistently setting up the ratio of the side lengths• confusing enlargements with reductions and vice versa


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August 10, 2015

Jul 1911:41 AM

Guided PracticeExample 1Is the transformation on the right a dilation? Justify your answer using the properties of dilations.

Steps1. Verify sides are parallel.

Jul 1911:59 AM

Example 3The following transformation represents a dilation. What is the scale factor? Does this indicate enlargement, reduction, or congruence?

Steps1. Verify shape, orientation, and angles have been preserved.2. Verify sides are parallel.3. Verify same scale, k.4. Verify vertices are collinear with the center, c.5. Draw conclusions.

Jul 1912:24 PM

IntroductionA figure is dilated if the preimage can be mapped to the image using a scale factor through a center point, usually the origin. You have been determining if figures have been dilated, but how do you create a dilation? If the dilation is centered about the origin, use the scale factor and multiply each coordinate in the figure by that scale factor. If a distance is given, multiply the distance by the scale factor.

Lesson 1.2

Jul 1912:25 PM

Key Concepts• The notation is as follows: Dk(x, y) = (kx, ky).• Multiply each coordinate of the figure by the scale factor when the center is at (0, 0).

• The lengths of each side in a figure also are multiplied by the scale factor.

• If you know the lengths of the preimage figure and the scale factor, you can calculate the lengths of the image by multiplying the preimage lengths by the scale factor.

• Remember that the dilation is an enlargement if k > 1, a reduction if 0 < k < 1, and a congruency transformation if k = 1.

Common Errors/Misconceptions• not applying the scale factor to both the x and ycoordinates in the point

• improperly converting the decimal from a percentage• missing the connection between the scale factor and the ratio of the image lengths to the preimage lengths

Jun 17:39 PM

Example 1If AB has a length of 3 units and is dilated by a scale factor of 2.25, what is the length of A'B'? Does this represent an enlargement or reduction?

Jul 1912:28 PM

Example 2A triangle has vertices G (2, –3), H (–6, 2), and J (0, 4). If the triangle is dilated by a scale factor of 0.5 through center C (0, 0), what are the image vertices? Draw the preimage and image on the coordinate plane.


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August 10, 2015

Jul 1912:33 PM

Example 3What are the side lengths of D'E'F' with a scale factor of 2.5 given the preimage and image to the right and the information that DE = 1, EF = 9.2, and FD = 8.6?

Jun 17:37 PM

IntroductionTwo basic instruments used in geometry are the straightedge and the compass. A straightedge is a bar or strip of wood, plastic, or metal that has at least one long edge of reliable straightness, similar to a ruler, but without any measurement markings. A compass is an instrument for creating circles or transferring measurements. It consists of two pointed branches joined at the top by a pivot. It is believed that during early geometry, all geometric figures were created using just a straightedge and a compass. Though technology and computers abound today to help us make sense of geometry problems, the straightedge and compass are still widely used to construct figures, or create precise geometric representations. Constructions allow you to draw accurate segments and angles, segment and angle bisectors, and parallel and perpendicular lines.

Lesson 2.1

Jun 18:04 PM

Words to know• altitude the perpendicular line from a vertex a figure to its opposite side; height

• bisect cut in half• congruent having the same shape, size, or angle• equidistant the same distance from a reference point• median of a triangle the segment joining the midpoints of two dies of a figure

Jun 18:04 PM

Example 1Copy the following segment using only a compass and a straightedge.

Jun 18:11 PM

Example 2Copy the following angle using only a compass and a straightedge.

Jun 18:04 PM

Example 3Use the line segment to construct a new line segment with length 2AB.


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Jun 18:11 PM

Example 4Use the given angle to construct a new angle equal to A + A (or 2A).

Jun 18:23 PM

IntroductionSegments and angles are often described with measurements. Segments have lengths that can be measured with a ruler. Angles have measures that can be determined by a protractor. It is possible to determine the midpoint of a segment. The midpoint is a point on the segment that divides it into two equal parts. When drawing the midpoint, you can measure the length of the segment and divide the length in half. When constructing the midpoint, you cannot use a ruler, but you can use a compass and a straightedge (or patty paper and a straightedge) to determine the midpoint of the segment. This procedure is called bisecting a segment. To bisect means to cut in half. It is also possible to bisect an angle, or cut an angle in half using the same construction tools. A midsegment is created when two midpoints of a figure are connected. A triangle has three midsegments.

Lesson 2.2

Jun 18:25 PM

Example 1Use a compass and straightedge to find the midpoint of PQ. Label the midpoint R.

Jun 18:25 PM

Example 2Construct a segment whose measure is 1/4 the length of PQ.

Jun 18:25 PM

Example 3Use a compass and straightedge to bisect an angle.

J

Jun 18:25 PM

Example 4Construct a angle whose measure is 3/4 the length of S.


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August 10, 2015

Jun 18:44 PM

Lesson 2.3IntroductionGeometry construction tools can also be used to create perpendicular and parallel lines. While performing each construction, it is important to remember that the only tools you are allowed to use are a compass and a straightedge, a reflective device and a straightedge, or patty paper and a straightedge. You may be tempted to measure angles or lengths, but in constructions this is not allowed. You can adjust the opening of your compass to verify that lengths are equal.

Jun 18:46 PM

Key ConceptsPerpendicular Lines and Bisectors• Perpendicular lines are two lines that intersect at a right angle (90˚).

A perpendicular line can be constructed through the midpoint of a segment. This line is called the perpendicular bisector of the line segment.

Jun 18:25 PM

Example 1Use a compass and straighedge to construct a perpendicular bisector of PQ.

Jun 18:25 PM

Example 2Use a compass and straighedge to construct a line perpendicular to line L through point A.

L

Jun 18:25 PM

Example 3Use a compass and straighedge to construct a line perpendicular to line m through point B that is not on the line.

Jun 18:25 PM

Example 4Use a compass and straighedge to construct a line parallel to line n through point C that is not on the line.


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August 10, 2015

Jul 1912:55 PM

IntroductionThe ability to copy and bisect angles and segments, as well as construct perpendicular and parallel lines, allows you to construct a variety of geometric figures, including triangles, squares, and hexagons. There are many ways to construct these figures and others. Sometimes the best way to learn how to construct a figure is to try on your own. You will likely discover different ways to construct the same figure and a way that is easiest for you. In this lesson, you will learn two methods for constructing a triangle within a circle.

Lesson 3.1

Jul 302:04 PM

Key ConceptsTriangles• A triangle is a polygon with three sides and three angles.• Triangles are classified based on their angle measure and the measure of their sides.

• Equilateral triangles are triangles with all three sides equal in length.

• The measure of each angle of an equilateral triangle is 60˚.Circles• A circle is the set of all points that are equidistant from a reference point, the center.

• The set of points forms a twodimensional curve that is 360˚.• Circles are named by their center. For example, if a circle has a center point, G, the circle is named circle G.

• The diameter of a circle is a straight line that goes through the center of a circle and connects two points on the circle. It is twice the radius.

• The radius of a circle is a line segment that runs from the center of a circle to a point on the circle.

• The radius of a circle is onehalf the length of the diameter.• There are 360˚ in every circle.Inscribing Figures• To inscribe means to draw a figure within another figure so that every vertex of the enclosed figure touches the outside figure.

• A figure inscribed within a circle is a figure drawn within a circle so that every vertex of the figure touches the circle.

• It is possible to inscribe a triangle within a circle. Like with all constructions, the only tools used to inscribe a figure are a straightedge and a compass, patty paper and a straightedge, reflective tools and a straightedge, or technology.

• This lesson will focus on constructions with a compass and a straightedge.

Jul 302:09 PM

Method 1: Constructing an Equilateral Triangle Inscribed in a Circle Using a Compass

1. To construct an equilateral triangle inscribed in a circle, first mark the location of the center point of the circle. Label the point X.

2. Construct a circle with the sharp point of the compass on the center point. 3. Label a point on the circle point A. 4. Without changing the compass setting, put the sharp point of the compass on A and

draw an arc to intersect the circle at two points. Label the points B and C. 5. Use a straightedge to construct BC. 6. (continued)7. Put the sharp point of the compass on point B. Open the compass until it extends to

the length of BC. Draw another arc that intersects the circle. Label the point D.8. Use a straightedge to construct BD and CD.9. Do not erase any of your markings.10. Triangle BCD is an equilateral triangle inscribed in circle X.

Method 2: Constructing an Equilateral Triangle Inscribed in a Circle Using a Compass

1. To construct an equilateral triangle inscribed in a circle, first mark the location of the center point of the circle. Label the point X.

2. Construct a circle with the sharp point of the compass on the center point. 3. Label a point on the circle point A. 4. Without changing the compass setting, put the sharp point of the compass on A and draw

an arc to intersect the circle at one point. Label the point of intersection B. 5. Put the sharp point of the compass on point B and draw an arc to intersect the circle at

one point. Label the point of intersection C.6. Continue around the circle, labeling points D, E, and F. Be sure not to change the

compass setting.7. Use a straightedge to connect A and C, C and E, and E and A.

Do not erase any of your markings.Triangle ACE is an equilateral triangle inscribed in circle X.

Common Errors/Misconceptions• inappropriately changing the compass setting• attempting to measure lengths and angles with rulers and protractors

• not creating large enough arcs to find the points of intersection

Jul 302:19 PM

Example 1Construct equilateral triangle ACE inscribed in circle O using Method 1.

Jul 302:19 PM

Example 2Construct equilateral triangle ACE inscribed in circle O using Method 2.

Jul 302:19 PM

Example 3Construct equilateral triangle JKL inscribed in circle P using Method 1. Use the length of HP as the radius for circle P.


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August 10, 2015

Jul 302:31 PM

Example 4Construct equilateral triangle JLN inscribed in circle P using Method 2. Use the length of as the radius for circle P.

Jul 302:39 PM

IntroductionTriangles are not the only figures that can be inscribed in a circle. It is also possible to inscribe other figures, such as squares. The process for inscribing a square in a circle uses previously learned skills, including constructing perpendicular bisectors.

Lesson 3.2

Jul 302:41 PM

Key Concepts• A square is a foursided regular polygon.• A regular polygon is a polygon that has all sides equal and all angles equal.

• The measure of each of the angles of a square is 90˚.• Sides that meet at one angle to create a 90˚ angle are perpendicular.

• By constructing the perpendicular bisector of a diameter of a circle, you can construct a square inscribed in a circle.

Constructing a Square Inscribed in a Circle Using a Compass1. To construct a square inscribed in a circle, first mark the location of the center point of the

circle. Label the point X. 2. Construct a circle with the sharp point of the compass on the center point. 3. Label a point on the circle point A. 4. Use a straightedge to connect point A and point X. Extend the line through the circle,

creating the diameter of the circle. Label the second point of intersection C. 5. Construct the perpendicular bisector of AC by putting the sharp point of your compass on

endpoint A. Open the compass wider than half the distance of AC. Make a large arc intersecting AC. Without changing your compass setting, put the sharp point of the compass on endpoint C. Make a second large arc. Use your straightedge to connect the points of intersection of the arcs.

6. Extend the bisector so it intersects the circle in two places. Label the points of intersection B and D.Use a straightedge to connect points A and B, B and C, C and D, and A and D.

Do not erase any of your markings.Quadrilateral ABCD is a square inscribed in circle X.

Common Errors/Misconceptions• inappropriately changing the compass setting• attempting to measure lengths and angles with rulers and protractors• not creating large enough arcs to find the points of intersection• not extending segments long enough to find the vertices of the square

Jul 302:50 PM

Example 1Construct square ABCD inscribed in circle O.

Jul 302:53 PM

Example 2Construct square JKLM inscribed in circle Q with the radius equal to onehalf the length of JL.

Jul 302:59 PM

IntroductionConstruction methods can also be used to construct figures in a circle. One figure that can be inscribed in a circle is a hexagon. Hexagons are polygons with six sides.

Lesson 3.3


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August 10, 2015

Jul 303:01 PM

Key Concepts• Regular hexagons have six equal sides and six angles, each measuring 120˚.

• The process for inscribing a regular hexagon in a circle is similar to that of inscribing equilateral triangles and squares in a circle.

• The construction of a regular hexagon is the result of the construction of two equilateral triangles inscribed in a circle.

Jul 303:06 PM

Common Errors/Misconceptions• inappropriately changing the compass setting• attempting to measure lengths and angles with rulers and protractors• not creating large enough arcs to find the points of intersection• not extending segments long enough to find the vertices of the hexagon

Jul 303:10 PM

Example 1Construct regular hexagon ABCDEF inscribed in circle O using Method 1.

Jul 303:13 PM

Example 2Construct regular hexagon ABCDEF inscribed in circle O using Method 2.

Jul 3111:11 AM

IntroductionThink about trying to move a drop of water across a flat surface. If you try to push the water droplet, it will smear, stretch, and transfer onto your finger. The water droplet, a liquid, is not rigid. Now think about moving a block of wood across the same flat surface. A block of wood is solid or rigid, meaning it maintains its shape and size when you move it. You can push the block and it will keep the same size and shape as it moves. In this lesson, we will examine rigid motions, which are transformations done to an object that maintain the object’s shape and size or its segment lengths and angle measures.

Lesson 4.1

Jul 3111:21 AM

Key Concepts• Rigid motions are transformations that don’t affect an object’s shape and size. This means that corresponding sides and corresponding angle measures are preserved.

• When angle measures and sides are preserved they are congruent, which means they have the same shape and size.

• The congruency symbol ( ) is used to show that two figures are congruent.

• The figure before the transformation is called the preimage.• The figure after the transformation is the image.• Corresponding sides are the sides of two figures that lie in the same position relative to the figure.

• Corresponding angles are the angles of two figures that lie in the same position relative to the figure. In transformations, the corresponding vertices are the preimage and image vertices, so A and A′ are corresponding vertices and so on.

• Transformations that are rigid motions are translations, reflections, and rotations.

• Transformations that are not rigid motions are dilations, vertical stretches or compressions, and horizontal stretches or compressions.


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Jul 3111:24 AM

Translating a Figure Given the Horizontal and Vertical Shift

1. Place your pencil on a vertex and count over horizontally the number of units the figure is to be translated.

2. Without lifting your pencil, count vertically the number of units the figure is to be translated. 3. Mark the image vertex on the coordinate plane. 4. Repeat this process for all vertices of the figure. 5. Connect the image vertices.

Translations• A translation is sometimes called a slide.• In a translation, the figure is moved horizontally and/or vertically.• The orientation of the figure remains the same.• Connecting the corresponding vertices of the preimage and image will result in a set of parallel lines.

Jun 238:29 PM

Reflections• A reflection creates a mirror image of the original figure over a reflection line.• Reflections are sometimes called flips.• The orientation of the figure is changed in a reflection.• In a reflection, the corresponding vertices of the preimage and image are equidistant from the line of reflection, meaning the distance from each vertex to the line of reflection is the same.• The line of reflection is the perpendicular bisector of the segments that connect the corresponding vertices of the preimage and the image.

Reflecting a Figure over a Given Reflection Line

Draw the reflection line on the same coordinate plane as the figure. If the reflection line is vertical, count the number of horizontal units one vertex is from the line and count the same number of units on the opposite side of the line. Place the image vertex there. Repeat this process for all vertices. If the reflection line is horizontal, count the number of vertical units one vertex is from the line and count the same number of units on the opposite side of the line. Place the image vertex there. Repeat this process for all vertices. If the reflection line is diagonal, draw lines from each vertex that are perpendicular to the reflection line extending beyond the line of reflection. Copy each segment from the vertex to the line of reflection onto the perpendicular line on the other side of the reflection line and mark the image vertices.Connect the image vertices.

Jul 3111:26 AM

Rotations• A rotation moves all points of a figure along a circular arc about a point. Rotations are sometimes called turns.

• In a rotation, the orientation is changed.• The point of rotation can lie on, inside, or outside the figure, and is the fixed location that the object is turned around.

• The angle of rotation is the measure of the angle created by the preimage vertex to the point of rotation to the image vertex. All of these angles are congruent when a figure is rotated.• Rotating a figure clockwise moves the figure in a circular arc about the point of rotation in the same direction that the hands move on a clock.

• Rotating a figure counterclockwise moves the figure in a circular arc about the point of rotation in the opposite direction that the hands move on a clock.

Rotating a Figure Given a Point and Angle of Rotation

1. Draw a line from one vertex to the point of rotation. 2. Measure the angle of rotation using a protractor. 3. Draw a ray from the point of rotation extending outward that creates the angle of

rotation. 4. Copy the segment connecting the point of rotation to the vertex (created in step 1) onto

the ray created in step 3. 5. Mark the endpoint of the copied segment that is not the point of rotation with the letter

of the corresponding vertex, followed by a prime mark (′ ). This is the first vertex of the rotated figure.

6. Repeat the process for each vertex of the figure. 7. Connect the vertices that have prime marks. This is the rotated figure.

Jul 3111:35 AM

Common Errors/Misconceptions• creating the angle of rotation in a clockwise direction instead of a counterclockwise direction and vice versa

• reflecting a figure about a line other than the one given• mistaking a rotation for a reflection• misidentifying a translation as a reflection or a rotation

Aug 261:12 PM

Example 1Describe the transformation that has taken place in the diagram below.

A

BCA'

B'C'

Jul 3111:37 AM

Example 2Describe the transformation that has taken place in the diagram to the right.


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August 10, 2015

Aug 261:08 PM

Example 3Describe the transformation that has taken place in the diagram below.

A

BC

RA' B'

C'

Jul 3111:41 AM

Example 4Rotate the given figure 45º counterclockwise about the origin.

Jul 3111:54 AM

IntroductionRigid motions can also be called congruency transformations. A congruency transformation moves a geometric figure but keeps the same size and shape. Preimages and images that are congruent are also said to be isometries. If a figure has undergone a rigid motion or a set of rigid motions, the preimage and image are congruent. When two figures are congruent, they have the same shape and size. Remember that rigid motions are translations, reflections, and rotations. Nonrigid motions are dilations, stretches, and compressions. Nonrigid motions are transformations done to a figure that change the figure’s shape and/or size.

Lesson 4.2

Jul 3111:57 AM

Key Concepts• To decide if two figures are congruent, determine if the original figure has undergone a rigid motion or set of rigid motions.

• If the figure has undergone only rigid motions (translations, reflections, or rotations), then the figures are congruent.

• If the figure has undergone any nonrigid motions (dilations, stretches, or compressions), then the figures are not congruent. A dilation uses a center point and a scale factor to either enlarge or reduce the figure. A dilation in which the figure becomes smaller can also be called a compression.

• A scale factor is a multiple of the lengths of the sides from one figure to the dilated figure. The scale factor remains constant in a dilation.

• If the scale factor is larger than 1, then the figure is enlarged.• If the scale factor is between 0 and 1, then the figure is reduced.• To calculate the scale factor, divide the length of the sides of the image by the lengths of the sides of the preimage.

• A vertical stretch or compression preserves the horizontal distance of a figure, but changes the vertical distance.

• A horizontal stretch or compression preserves the vertical distance of a figure, but changes the horizontal distance.

• To verify if a figure has undergone a nonrigid motion, compare the lengths of the sides of the figure. If the sides remain congruent, only rigid motions have been performed.

• If the side lengths of a figure have changed, nonrigid motions have occurred.

Common Errors/Misconceptions• mistaking a nonrigid motion for a rigid motion and vice versa• not recognizing that rigid motions preserve shape and size• not recognizing that it takes only one nonrigid motion to render two figures not congruent

Jul 3111:59 AM

Example 1Determine if the two figures to the right are congruent by identifying the transformations that havetaken place.

Jun 238:50 PM

Example 2Determine if the two figuresto the right are congruent by identifying the transformations that havetaken place.


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August 10, 2015

Jul 3111:54 AM

IntroductionIf a rigid motion or a series of rigid motions, including translations, rotations, or reflections, is performed on a triangle, then the transformed triangle is congruent to the original. When two triangles are congruent, the corresponding angles have the same measures and the corresponding sides have the same lengths. It is possible to determine whether triangles are congruent based on the angle measures and lengths of the sides of the triangles.

Lesson 5.1

Jun 239:00 PM

Key Concepts• To determine whether two triangles are congruent, you must observe the angle measures and side lengths of the triangles.

• Corresponding angles are a pair of angles in a similar position.• If two or more triangles are proven congruent, then all of their corresponding parts are congruent as well. This postulate is known as Corresponding Parts of Congruent Triangles are Congruent (CPCTC). A postulate is a true statement that does not require a proof.

Common Errors/Misconceptions• incorrectly identifying corresponding parts of triangles• assuming corresponding parts indicate congruent parts• assuming alphabetical order indicates congruence

Jun 239:03 PM

Example 1Use corresponding parts to identify the congruent triangles.

Jun 239:04 PM

Example 2

Name the corresponding angles and sides of the congruent triangles.

Aug 281:50 PM

Use construction tools to determine if the triangles are congruent. If they are, name the congruent triangles and corresponding angles and sides.

Jul 3111:54 AM

IntroductionWhen a series of rigid motions is performed on a triangle, the result is a congruent triangle. When triangles are congruent, the corresponding parts of the triangles are also congruent. It is also true that if the corresponding parts of two triangles are congruent, then the triangles are congruent. It is possible to determine if triangles are congruent by measuring and comparing each angle and side, but this can take time. There is a set of congruence criteria that lets us determine whether triangles are congruent with less information.

Lesson 5.2


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August 10, 2015

Jun 239:14 PM

Key Concepts• The SideSideSide (SSS) Congruence Statement states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

• The SideAngleSide (SAS) Congruence Statement states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

• The included angle is the angle that is between the two congruent sides.

Included side Nonincluded side

is included between ∠C and ∠A.is included between ∠F and ∠D.

is NOT included between ∠C and ∠A.is NOT included between ∠F and ∠D.

Jun 239:17 PM

SideSideSide (SSS) SideAngleSide (SAS) AngleSideAngle (ASA)

Jun 239:23 PM

Example 1Determine which congruence statement, if any, can be used to show that and are congruent.

Jun 239:25 PM

Example 2Determine which congruence statement, if any, can be used to show that and are congruent.

Jul 288:53 AM

Lesson 6.1

IntroductionCongruent triangles have corresponding parts with angle measures that are the same side and lengths that are the same. If two triangles are congruent, they are also similar. Similar triangles have the same shape, but may be different in size. It is possible for two triangles to be similar but not congruent. Just like with determining congruency, it is possible to determine similarity based on the angle measures and lengths of the sides of the triangles.

Jul 2810:30 AM

Key Concepts• When a triangle is transformed by a similarity transformation (a rigid motion [reflection, translation, or rotation] followed by a dilation), the result is a triangle with a different position and size, but the same shape.

• If two triangles are similar, then their corresponding angles are congruent and the measures of their corresponding sides are proportional, or have a constant ratio.

• The ratio of corresponding sides is known as the ratio of similitude.

• The scale factor of the dilation is equal to the ratio of similitude.• Similar triangles with a scale factor of 1 are congruent triangles.


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Jul 2810:33 AM

Example 1Use the definition of similarity in terms of similarity transformations to determine whether the two figures are similar. Explain your answer.

Jul 2810:34 AM

Example 2Use the definition of similarity in terms of similarity transformations to determine whether the two figures are similar. Explain your answer.

Aug 103:39 PM

Example 3A dilation of centered at point P with a scale factor of 2 is then reflected over the line L. Determine if is simlar to . If possible, find the unknown angle measures and lengths in .

D

E

T U3

5

F

V

Lesson 6.2

Lesson 6.2

Key Concepts• The AngleAngle (AA) Similarity Statement is one statement that allows us to prove triangles are similar.

• The AA Similarity Statement allows that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Aug 129:35 AM

Example 1Explain why the triangles are similar and write a similarity statement.

A

BC

D

E

Aug 129:25 AM

Example 2Explain why , and then find the length of DF.


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August 10, 2015

Aug 129:37 AM

Example 3Identify the similar triangles. Find x and the measures of the indicated sides.

A

C

BG

H

Jx+1

x+5

3

6

Aug 129:34 AM

Example 4Suppose a person 5 feet 10 inches tall casts a shadow that is 3 feet 6 inches long. At the same time of day, a flagpole casts a shadow that is 12 feet long. To the nearest foot, how tall is the flagpole?

Aug 129:40 AM

Lesson 7.1

IntroductionThere are many ways to show that two triangles are similar, just as there are many ways to show that two triangles are congruent. The AngleAngle (AA) Similarity Statement is one of them. The SideAngleSide (SAS) and SideSideSide (SSS) similarity statements are two more ways to show that triangles are similar. In this lesson, we will prove that triangles are similar using the similarity statements.

Aug 129:45 AM

Key Concepts• The SideAngleSide (SAS) Similarity Statement asserts that if the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.

• The SideSideSide (SSS) Similarity Statement asserts that if the measures of the corresponding sides of two triangles are proportional, then the triangles are similar.

• Similar triangles have corresponding sides that are proportional, whereas congruent triangles have corresponding sides that are of the same length.

Aug 129:49 AM

Example 1Prove

A

B

C D

E

4 6

812

9 6

Aug 129:48 AM

Example 2Determine whether the triangles are similar. Explain your reasoning.


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August 10, 2015

Aug 129:49 AM

Example 3Determine whether the triangles are similar. Explain your reasoning.

Aug 129:49 AM

Example 4Identify similar triangles and then find the value of x.

A

B

C

DEF

3

2 2.5 x

6

Aug 1210:26 AM

Lesson 7.2

IntroductionArchaeologists, among others, rely on the AngleAngle (AA), SideAngleSide (SAS), and SideSideSide (SSS) similarity statements to determine actual distances and locations created by similar triangles. Many engineers, surveyors, and designers use these statements along with other properties of similar triangles in their daily work. Having the ability to determine if two triangles are similar allows us to solve many problems where it is necessary to find segment lengths of triangles.

Aug 1210:42 AM

Triangle Proportionality TheoremIf a line parallel to one side of a triangle intersects the other two sides of the triangle, then the parallel line divides these two sides proportionally.

• In the figure above, ; therefore, .

Aug 1210:52 AM

Triangle Angle Bisector TheoremIf one angle of a triangle is bisected, or cut in half, then the angle bisector of the triangle divides the opposite side of the triangle into two segments that are proportional to the other two sides of the triangle.

• In the figure above, ; therefore, .

Aug 1210:56 AM

Example 1Find the length of BE.


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Aug 1210:56 AM

Example 2Using two different methods, find the length of CA.

A

B

CD

E

9

6

5

Aug 1210:56 AM

Example 3Prove that .

A

B

C

DE

9 10.5

3.53

Aug 1210:56 AM

Example 4Is ?

Aug 189:19 AM

Example 5Find the lengths of BD and DC.

A

B

C

D

7x+2

x+1

5.6

Aug 1211:08 AM

Lesson 7.3

Corresponding sides of similar triangles are proportional; therefore, and .

Aug 1211:15 AM

Example 1Write a twocolumn proof to prove the Pythagorean Theorem using similar triangles.

A

BC a

bc

Statements Reasons1.

2.

3.

4.

5.

6.

7.

8.


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Aug 1211:15 AM

Example 2Find the length of the altitude, x, of .

Aug 1211:15 AM

Example 3Find the unknown values in the figure.

Aug 1211:23 AM

Lesson 7.4

IntroductionDesign, architecture, carpentry, surveillance, and many other fields rely on an understanding of the properties of similar triangles. Being able to determine if triangles are similar and understanding their properties can help you solve realworld problems.

Aug 1211:26 AM

Example 1A meterstick casts a shadow 65 centimeters long. At the same time, a tree casts a shadow 2.6 meters long. How tall is the tree?

Aug 1211:28 AM

Example 2Finding the distance across a canyon can often be difficult. A drawing of similar triangles can be used to make this task easier. Use the diagram to determine AR, the distance across the canyon.

A

R

C

D

B

x

180 m90 m

75 m

Aug 1211:27 AM

Example 3To find the distance across a pond, Rita climbs a 30foot observation tower on the shore of the pond and locates points A and B so that AC is perpendicular to CB. She then finds the measure of DB to be 12 feet. What is the measure of AD, the distance across the pond?

A B

C

x

30

12D


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Aug 1211:32 AM

Example 4To estimate the height of an overhang, a surveyor positions herself so that her line of sight to the top of the overhang and her line of sight to the bottom form a right angle. What is the height of the overhang to the nearest tenth of a meter?

A

B

CD8.5 m

1.75 m

Aug 1211:36 AM

Lesson 8.1

Key Concepts• Adjacent angles are angles that lie in the same plane and share a vertex and a common side. They have no common interior points.

• Vertical Angles Theorem Vertical angles are congruent. ABC and EBD are vertical angles.

ABE and CBD are vertical angles

Aug 1212:11 PM

Example 1Look at the following diagram. List pairs of supplementary angles, pairs of vertical angles, and a pair of opposite rays.

AB

C

D

E

F

12 345 6

7 8

Aug 1212:11 PM

Example 2Prove the theorem that angles complementary to congruent angles are congruent using the given information.

In the figure below, prove that is congruent to , given that is perpendicular to and is congruent to .

ABC

D

12 3

4

Aug 1212:11 PM

Example 3In the diagram below, AC and BD are intersecting lines. If and , find and .

A B

CD

12

34

Aug 1212:11 PM

Example 4Prove that vertical angles are congruent given a pair of intersecting lines, and .


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August 10, 2015

Aug 1212:12 PM

Example 5In the diagram, DB is the perpendicular bisector of AC. If AD = 4x – 1 and DC = x + 11, what are the values of AD and DC ?

Aug 1212:41 PM

Key Concepts• The interior angles lie between the parallel lines and the exterior angles lie outside the pair of parallel lines. In the following diagram, line k is the transversal. A transversal is a line that intersects a system of two or more lines.

Lesson 8.2

Aug 1212:52 PM

Corresponding Angles PostulateIf two parallel lines are cut by a transversal, then corresponding angles are congruent.Alternate Interior Angles TheoremIf two parallel lines are intersected by a transversal, then alternate interior angles are congruent.SameSide Interior Angles TheoremIf two parallel lines are intersected by a transversal, then sameside interior angles are supplementary.SameSide Exterior Angles TheoremIf two parallel lines are intersected by a transversal, then sameside exterior angles are supplementary.

Aug 218:32 AM

Example 1Given , prove that .

A B

C

D E

Aug 1212:54 PM

Example 2Given two parallel lines and a transversal, prove that alternate interior angles are congruent. In the following diagram, line l and m are parallel. Line k is the transversal.

l

m

k

1 23 4

5 67 8

Aug 1212:54 PM

Example 3In the diagram, and . If , , and , find the measures of the unknown angles and the values of x and y.


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Aug 1212:58 PM

Example 4In the diagram, . If m 1 = 35 and m 2 = 65, find m EQF.

Aug 218:42 AM

Lesson 9.1

Key Concepts• All of the angles of an acute triangle are acute, or less than 90˚.• One angle of an obtuse triangle is obtuse, or greater than 90˚. • A right triangle has one angle that measures 90˚.• A scalene triangle has no congruent sides.• An isosceles triangle has at least two congruent sides.• An equilateral triangle has three congruent sides.

Aug 218:42 AM

Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.

m D = m A + m B

Aug 218:42 AM

Example 1Find the measure of C.

Aug 218:42 AM

Example 2Find the missing angle measures.

Aug 218:42 AM

Example 3Find the missing angle measures.

3x10

4x+60155


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August 10, 2015

Aug 258:40 AM

Example 4The Triangle Sum Theorem states that the sum of the angle measures of a triangle is 180. Write a twocolumn proof for this theorem.

A

B

C1

2

3

Statements Reasons1.2.3.4.5.6.7.8.

Aug 258:38 AM

Lesson 9.2

Aug 258:46 AM

Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the congruent sides are congruent.

Aug 258:44 AM

Example 1Find the measure of each angle of .

A

B C(4x) (6x36)

Aug 258:44 AM

Example 2Determine whether with vertices A (–4, 5), B (–1, –4), and C (5, 2) is an isosceles triangle. If it is isosceles, name a pair of congruent angles.

Aug 258:46 AM

Example 3Given , prove that . A

B C

Statements Reasons1.2.3.4.5.6.


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August 10, 2015

Aug 258:45 AM

Example 4Find the values of x and y.

Aug 258:53 AM

Example 5 is equilateral. Prove that it is equiangular.

Statements Reasons1.

2.

3.

4.

5.6.7.

Aug 258:39 AM

Lesson 9.3

Key Concepts• The midpoint is the point on a line segment that divides the segment into two equal parts.

• A midsegment of a triangle is a line segment that joins the midpoints of two sides of a triangle.

Aug 259:30 AM

Triangle Midsegment Theorem A midsegment of a triangle is parallel to the third side and is half as long.

Aug 259:31 AM

Key Concepts• Every triangle has three midsegments.

Aug 259:32 AM

Example 1Find the lengths of BC and YZ and the measure of AXZ.


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Aug 259:32 AM

Example 2If AB = 2x+7 and YZ = 3x 6.5, what is the length of AB?

(2x+7)

3x6.5

A

B

C

X

Y

Z

Aug 259:32 AM

Example 3The midpoints of a triangle are X (–2, 5), Y (3, 1), and Z (4, 8). Find the coordinates of the vertices of the triangle.

Aug 259:33 AM

Example 4Write a coordinate proof of the Triangle Midsegment Theorem using the graph below.

A (2a,2b)

D E

B (0,0) C (2c,0)

Aug 218:53 AM

Lesson 9.4Key Concepts• Every triangle has four centers. • Each center is determined by a different point of concurrency—the point at which three or more lines intersect.

• These centers are the circumcenter, the incenter, the orthocenter, and the centroid.

Aug 259:42 AM

People be crazy Perpendicular Bisector circumcenter

A bit insane Angle Bisector incenter

Me and 50 cent Median centroid

All too vain Altitude orthocenter

Aug 2511:35 AM

Circumcenter TheoremThe circumcenter of a triangle is equidistant from the vertices of a triangle.

The circumcenter of this triangle is at X.


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August 10, 2015

Aug 251:18 PM

Centroid Theorem The centroid of a triangle isthe distance from eachvertex to the midpoint of the opposite side.

The centroid of this triangle is at point X.

Aug 251:25 PM

Example 1 has vertices A (3,3), B (7,3), and C (3,3). Justify that (5,0) is the circumcenter of .

Aug 251:25 PM

Example 2 has vertices A (1,6), B (7,6), and C (3,2). Find the equation of each altitude of to verify that (3,4) is the orthocenter of .

Aug 251:22 PM

Example 3 has vertices A (–2, 4), B (5, 4), and C (3, –2). Find the equation of each median of to verify that (2, 2) is the centroid of .

Aug 251:24 PM

Example 4Using from Example 3, which has vertices A (–2, 4), B (5, 4), and C (3, –2), verify that the centroid, X (2, 2), is the distance from each vertex.

Aug 218:53 AM

Lesson 10.1

Key Concepts• A convex polygon is a polygon with no interior angle greater than 180º and all diagonals lie inside the polygon.

• A concave polygon is a polygon with at least one interior angle greater than 180º and at least one diagonal that does not lie entirely inside the polygon.

• A parallelogram is a special type of quadrilateral with two pairs of opposite sides that are parallel.


25

August 10, 2015

Aug 268:47 AM

Example 1Quadrilateral ABCD has the following vertices: A (–4, 4), B (2, 8), C (3, 4), and D (–3, 0). Determine whether the quadrilateral is a parallelogram. Verify your answer using slope and distance to prove or disprove that opposite sides are parallel and opposite sides are congruent.

Aug 268:50 AM

Example 2Use the parallelogram from Example 1 to verify that the opposite angles in a parallelogram are congruent and consecutive angles are supplementary given that and .

Sep 29:01 AM

Example 3Use the parallelogram from Example 1 to prove that diagonals of a parallelogram bisect each other.

Sep 29:04 AM

Example 4Use the parallelogram from Example 1 and the diagonal DB to prove that a diagonal of a parallelogram separates the parallelogram in two congruent segments.

Aug 218:42 AM

Lesson 10.2Key ConceptsIf a parallelogram is a rectangle, then the diagonals are congruent.

Sep 211:37 AM

Key Concepts• A rhombus is a special parallelogram with all four sides congruent.

If a parallelogram is a rhombus, the diagonals of the rhombus bisect the opposite pairs of angles.


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August 10, 2015

Sep 212:09 PM

Properties of Trapezoids

BA and CD are the legs.BC and AD are the bases.

Sep 212:11 PM

Isosceles Trapezoid

Sep 212:12 PM

Properties of Kites

Sep 212:12 PM

Hierarchy of Quadrilaterals

Sep 212:13 PM

Example 1Quadrilateral ABCD has vertices A (–6, 8), B (2, 2), C (–1, –2), and D (–9, 4). Using slope, distance, and/or midpoints, classify as a rectangle, rhombus, square, trapezoid, isosceles trapezoid, or kite.

Sep 212:14 PM

Example 2Quadrilateral ABCD has vertices A (0, 8), B (11, 1), C (0, –6), and D (–11, 1). Using slope, distance, and/or midpoints, classify as a rectangle, rhombus, square, trapezoid, isosceles trapezoid, or kite.


27

August 10, 2015

Sep 212:15 PM

Example 3Quadrilateral ABCD has vertices A (1, 2), B (1, 5), C (4, 3), and D (2, 0). Using slope, distance, and/or midpoints, classify as a rectangle, rhombus, square, trapezoid, or kite.

Sep 212:15 PM

Example 4Use what you know about the diagonals of rectangles, rhombuses, squares, kites, and trapezoids to classify the quadrilateral given the vertices M (0, 3), A (5, 2), T (6, 3), and H (1, 4).

Sep 212:15 PM

Example 5Use what you know about the diagonals of rectangles, rhombuses, squares, kites, and trapezoids to classify the quadrilateral given the vertices P (1, 5), Q (5, 2), R (4, 3), and S (4, 3).

Sep 212:19 PM

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